Geometric Averaging and Deterministic Limits for Stochastically Renormalized Field Equations

We present a general averaging theorem showing that a family of stochastically renormalized field equations parameterized by a small scale parameter (which goes to zero) converges (in probability) to a deterministic effective PDE obtained by geometric averaging under scale-separation and ergodicity hypotheses. The main result gives explicit convergence modes and error bounds under mixing/ergodicity and regularity of the renormalization maps. Techniques used combine stochastic homogenization, two-scale convergence, martingale methods and compactness arguments familiar from SPDE limits. A toy stochastic Maxwell-like PDE on the 2-torus (T2) is worked out formally, and a numerical scheme outline (HMM/multiscale FEM plus Monte Carlo sampling) is provided.